A Domain Wall Solution by Perturbation of the Kasner Spacetime

Home , Kasner metric

arXiv:1201.5362v1 [gr-qc] 25 Jan 2012 oanWl ouinb etraino h anrSpaceti Kasner the of Perturbation by Solution Wall Domain A et fPyia n niomna cecs nvriyo University Sciences, Environmental and Physical of Dept. aur 4 2012 24, January et fAtooyadAtohsc,Uiest fToronto of University Astrophysics, and Astronomy of Dept. Received 9633FotS.Ws,Trno NMV3R7 M5V ON Toronto, West, St. Front 1906-373 [email protected] [email protected] ereKotsopoulos George hre .Dyer C. Charles accepted ; and oot croog,and Scarborough, Toronto f oot,Canada Toronto, , me –2–

Abstract

Plane symmetric perturbations are applied to an axially symmetric Kasner spacetime which leads to no momentum ﬂow orthogonal to the planes of symmetry. This ﬂow appears laminar and the structure can be interpreted as a domain wall. We further extend consideration to the class of Bianchi Type I spacetimes and obtain corresponding results.

Subject headings: general relativity, Kasner metric, Bianchi Type I, domain wall, perturbation –3–

1. Finding New Solutions

There are three principal exact solutions to the Einstein Field Equations (EFE) that are most relevant for the description of astrophysical phenomena. They are the Schwarzschild (internal and external), Kerr and Friedman-LeMaître-Robertson-Walker (FLRW) models. These solutions are all highly idealized and involve the introduction of simplifying assumptions such as symmetry.

The technique we use which has led to a solution of the EFE involves the perturbation of an already symmetric solution (Wilson and Dyer 2007). Whereas standard perturbation techniques involve specifying the forms of the energy-momentum tensor to determine the form of the metric, we begin by deﬁning a new metric with an applied perturbation:

gab =g ˜ab + hab (1)

where g˜ab is a known symmetric metric and hab is the applied perturbation. We then use this new metric to ﬁnd the components of the Einstein tensor, Gab, in terms of the metric components. Next, we invert the EFE, where now Tab = Gab/κ, to ascertain whether or not

components of the energy-momentum tensor, Tab, exist to satisfy the perturbation. If such an energy-momentum tensor is physically acceptable, we then try to limit the behaviour of our perturbation by imposing conditions to model particular astronomical phenomenon. For example, if we were modelling a galaxy-like structure, we could choose our density to scale as ρ r−2 from the galactic centre (Wilson and Dyer 2007). This makes the ∼ technique ﬂexible and relevant in that we can model a physically viable structure. With these real physical constraints applied, we have a new metric. We now re-compute the energy-momentum components of this new metric and investigate any implications that may arise from our new exact solution to the EFE. –4–

2. The Base Solution

The metric we choose to perturb is the Kasner spacetime (Kasner 1921). This metric describes an anisotropic universe that is a vacuum solution to the EFE classiﬁed as a Bianchi Type I universe. It is characterized by a spacetime that is anisotropically expanding (or contracting) in two directions while contracting (or expanding) in the third with space-like slices that are spatially ﬂat and with a singularity occurring at t = 0. Hence, the Kasner spacetime is of special interest in cosmology since the standard cosmological solutions to the EFE near the cosmological singularity, such as the aforementioned FLRW model, can be described as a succession of Kasner epochs (Lifshitz and Khalatnikov 1963). Furthermore, by breaking a homogeneity along one direction, Bianchi Type solutions can

easily be generalized to spacetimes with G2 isotropy groups (Harvey 1990).

The Kasner spacetime in Cartesian coordinates with metric signature (+, , ..., ) − − takes the form (in spacetime dimension D):

D−1 2 2 2 2 ds = dt t Pj [dx ] (2) − j j=1 X

where Pj are the Kasner exponents which satisfy:

D−1 D−1 2 Pj =1 and Pj =1 (3) j=1 j=1 X X The ﬁrst condition in (3) describes a plane whereas the second condition describes a sphere of dimension D 1. Thus, the Kasner exponents lie on a sphere of dimension D 2. For − − D = 4, at each t = constant hypersurface, there exists a ﬂat 3-dimensional space, whose worldlines of constant x, y and z are time-like geodesics along which galaxies, or other test particles, can be imagined to move. Since the solution represents an anisotropically expanding (or contracting) universe, a volume element dV increases (or decreases) in time as g d3x = td3x, where g = det g . | | | ab| p –5–

The Kasner spacetime in Cartesian coordinates in a 4 D spacetime takes the form: −

ds2 = dt2 t2P1 dx2 t2P2 dy2 t2P3 dz2 (4) − − − with the same restrictions imposed by the Kasner exponents as given in (3).

The form of the Kasner spacetime that we consider is the axisymmetric case with G2 isometry that occurs when two of the exponents are equal, which by (3), requires that

P1 = P2 =2/3 and P3 = 1/3. With this restriction, the metric in cylindrical coordinates − becomes: ds2 = dt2 t4/3(dr2 + r2dφ2) t−2/3dz2 (5) − −

3. The Perturbations

The set of perturbations we implement are only on the z-component of the Kasner spacetime represented in cylindrical coordinates (5). Clearly, the results we obtain in cylindrical coordinates will be similar to those in Cartesian coordinates since r2 = x2 + y2. For simplicity, we deﬁne the Kasner spacetime in cylindrical coordinates as:

ds2 = dt2 t4/3(dr2 + r2dφ2) g dz2 (6) − − zz where gzz =g ˜zz + hab is our perturbed metric. We consider the three Cases shown in Table 1.

Cases gzz =g ˜zz + hab

I [t−2/3 + H] II [t−1/3 + H]2 III [t−2/3 + Z(z)T (t)]

Table 1: Implemented perturbations. –6–

In Cases I and II, we first investigate H as a differentiable function of z. We then investigate Cases I and II again, but where H is now made a differentiable function of t, r and z. In Case III, we investigate the result of a perturbation with a product separable differentiable solution.

To aid in these computations, we use the REDUCE Computer Algebra System (Hearn 2009) with the REDTEN Tensor Analysis Package (Harper and Dyer 1994).

4. Results of the Perturbations

Both Cases I and II produce a Gab with non-zero components. When the perturbation is H(z), the result produces non-zero terms for Gtt, Grr and Gφφ. All other terms, including

Gzz, are zero. When the perturbation is H(t, r, z), a similar result is obtained for all cases, but with the addition of cross-components, G for a = b. These cross terms can ab 6 be made zero if we choose the perturbation to contain only ﬁrst-order terms in H or by making ∂H/∂r = ∂H/∂φ = 0. Also, in cylindrical coordinates for all the cases reviewed, 2 Grr = r Gφφ, which is expected. The product separable perturbation of Case III reveals

the same results as Case I and II; we arrive at non-zero terms for Gtt, Grr and Gφφ and a

Gzz =0. Thus all perturbations that involved only the gzz term of the Kasner metric reveal that we will always obtain Gzz = 0 when considering axisymmetric or plane symmetric Kasner spacetimes, when P 1= P 2.

Relating these results to the energy-momentum tensor Tab, both the Gzz and Gza = Gaz (for a = 0, 1, 2 ) components of the Einstein tensor being zero reveals that there is no { } momentum-ﬂux across the z = constant surface regardless of the type of perturbation applied. Each of the applied perturbations resulted in the metric collapsing (or expanding) in the z-direction while expanding (or collapsing) in the x- and y-directions, as is expected –7–

of the Kasner metric. But, as a result of the Gzz term being zero, there is no interaction between the stratiﬁed layers of the matter above or below the xy-plane. This is suggestive of laminar ﬂow.

Inverting the EFE to get Tab = Gab/κ, leads to Tab being diagonal but with no Tzz component. The energy-momentum tensor for an inﬁnite, static plane-symmetric domain wall as suggested by Campanelli et al is:

T = δ(z)diag(ρ, p, p, 0) (7) ab − − where ρ is the energy-density and p is the pressure. This energy-momentum tensor (Campanelli et al. 2003) corresponds to an inﬁnite, static plane-symmetric domain wall (Kibble 1976) lying in the xy-plane. Domain walls correspond to a particular class of topological defects whereby the energy-density is trapped, and from a cosmological point of view, these over-dense regions could lead to structure formation (Brandenberger 1997). Furthermore, it has been suggested (Friedland et al. 2003) that the Universe may be dominated by a network of domain walls and these domain wall structures could represent an alternative view of dark energy theories.

5. Energy Conditions

In order to rule out any non-physical solutions to the EFE, energy conditions are applied to the state of matter content for gravitational and non-gravitational ﬁelds. These energy conditions consist of the Weak, Null, Strong and Dominant energy conditions, which are coordinate-invariant constraints on the energy-momentum tensor.

For the cases in which our Kasner spacetime was perturbed with H(z), we apply the least stringent of these conditions, the Null Energy Condition (NEC). This condition states –8– that for all future-pointing null vectors, ka:

a b Tabk k ≧ 0 (8)

This restriction implies that ρ + p ≧ 0, whereby the energy-density may be negative as long as there is a compensating pressure. If this condition is violated in any of the perturbed Kasner spacetimes, it would then indicate that our solutions are unstable.

We choose the null vector ﬁeld ka, in the Kasner spacetime to be,

2 1 n g 2 ka = − zz , 0, 0, n (9) s gtt !

where n ǫ R. This class of null vectors lies in the Tzz plane of interest. A summary of the energy-density restrictions is shown in Table 2, where T˙ = ∂T/∂t. For all Cases I to III,

a b Tabk k is positive deﬁnite for positive H.

a b Cases gzz =˜gzz + hab Tabk k 1 3 1 3 2 4t / (t / n +t)H I [t−2/3 + H(z)] 9t4/3(6t2/3H+1) 1 3 2 4(t / n +t)H II [t−1/3 + H(z)]2 9t4/3(t1/3H+1) 1 3 1 3 2 2 1 3 2 2t / (3t / Tn˙ t+3T˙ t +2t / T n +2T t)H III t−2/3 + H(z)T (t) 9t4/3(6t2/3HT +1)

Table 2: Applied NEC results

6. Bianchi Type I Metrics

As mentioned in Section 2, the Kasner spacetime is a special class of Bianchi Type I spacetime being a homogeneous and anisotropic vacuum solution to the EFE. Therefore, it is appropriate to apply the perturbations to the more generalized Bianchi Type I to –9– investigate its dynamics and compare the results to the perturbed Kasner models. The Bianchi Type I (Stephani 2003) metric has the form:

2 2 2 2 2 2 ds = c dt g11dx g22dy g33dz (10) − − −

where, 1 3 2 2 3 g =( g) / [ct/(Mctˆ + A)] Pα− / (11) αα − such that α = 1, 2, 3, with no sum on α. The quantity Mˆ = κµc2√ g is a constant and − √ g =3ct(Mctˆ + A)/4, where A is an integration constant. Using these relationships and − an appropriate constant rescaling of coordinates, we can re-write (10) as:

ds2 = dt2 t4/3dx2 t4/3dy2 t−2/3(1 + ǫt)2dz2 (12) − − − where ǫ = M/Aˆ . As before, since we wish to compare this result to the axisymmetric perturbed Kasner spacetime, we impose P1 = P2 =2/3 and P3 = 1/3. − There are two limiting cases of the Bianchi Type I metric in (12). If we choose ǫt 1, ≪ the initial singularity is approached at early times and we regain the original Kasner spacetime. For late times, if we choose ǫt 1, the metric isotropizes to the well-known ≫ Einstein-de Sitter dust metric, which is a sub-set of the FLRW metric. Indeed, the Kasner solution is a past asymptotic state as mentioned in Section 2.

7. Perturbations on Bianchi Type I

The perturbations applied to the Bianchi Type I metric are summarized in Table 3. – 10 –

Cases gzz =g ˜zz + hab

I t−2/3[(1 + ǫt)2 + H(z)] II t−2/3[(1 + ǫt)2 + H(z)T (t)]

Table 3: Implemented Perturbations on Bianchi Type I

Investigating Gab for the perturbations of Cases I and II, we arrive at the same conclusion

as was found for our perturbed Kasner spacetimes; we obtain Gzz =0 for the axisymmetric Bianchi Type I spacetimes.

8. Symmetries

The nonlinear nature of the EFE makes it diﬃcult to ﬁnd exact solutions. All of the known solutions have admitted simplifying symmetries in order to attain a solution. The technique we apply involves the perturbation of an already highly symmetric spacetime in hopes of producing a new spacetime solution. We now wish to determine to what extent any symmetries that were inherent in the original spacetime metric still retain (or break) any symmetries.

To investigate symmetries, we will consider conformal Killing vector ﬁelds, ξa, that satisfy the conformal Killing equations:

ξa||b + ξb||a = φgab (13)

where φ = 1 ξc and denotes covariant diﬀerentiation. Utilizing the REDUCE/REDTEN 2 ||c || computer algebra system, we computed the conformal Killing equations as 10 symmetric rank-2 tensors for each of the perturbation cases investigated. A Killing vector from the original metric was then calculated along with its covariant derivative and substituted in the conformal Killing equation (13). For the perturbations applied to the two Cases, no – 11 – such Killing vectors or conformal Killing vectors were found to remain; the original Killing vector is no longer a Killing vector.

9. Conclusion

Starting from two known, highly symmetric, solutions to the Einstein Field Equations we have applied plane symmetric perturbations and have shown that the resulting perturbed spacetimes exhibit the existence of structure that can be interpreted as a domain wall. It was demonstrated that these solutions do not violate the Null Energy Condition and thus would permit us to apply an appropriate energy-momentum tensor in future investigations.

10. Acknowledgements

This research was supported in part by the Natural Sciences and Engineering Research Council of Canada via a Discovery Grant to CCD. – 12 –

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